最小方形铺砌的稀疏角差MILP
A Sparse Corner-Difference MILP for Minimum Square Tiling
- Shenzhen Research Institute of Big Data（深圳大数据研究院）
- The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳））
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
针对最小方形铺砌问题，提出角差MILP（CD-MILP），将非零系数从Θ(N^5)降至Θ(N^3)，并引入角排序约束处理对称性，实验表明组合方法显著提升求解性能。
AI中文摘要：
我们研究用边长为严格小于N的整数的轴对齐正方形铺砌一个N×N正方形的问题，目标是最小化瓷砖数量。标准的基于放置的精确覆盖MILP包含Θ(N^5)个放置到单元格的非零系数，并且还受到正方形域的二面体对称性的影响。我们引入了一种角差MILP（CD-MILP），它通过至多四个带符号的角系数来表示每个选定的正方形。二维前缀重建表明，由此产生的约束等价于原始的单元格覆盖方程，同时将与覆盖相关的非零数量减少到Θ(N^3)。我们还引入了轻量级的角排序约束，这些约束从每个D4轨道中至少保留一个代表，尽管平局可能留下残余对称性。在九个素数大小的实例上的计算实验比较了基线、CD-MILP、带角排序的基线以及它们的组合，使用了Gurobi和COPT求解器。在所述的实验设置下，组合公式用Gurobi解决了八个实例，用COPT解决了七个，而基线分别解决了四个和五个。其平均预求解矩阵平均包含约10^5个非零系数，而基线约为10^7个。
英文摘要：
We study the problem of tiling an $N\times N$ square with axis-aligned squares whose side lengths are integers strictly less than $N$, with the objective of minimizing the number of tiles. The standard placement-based exact-cover MILP contains $Θ(N^5)$ placement-to-cell nonzero coefficients and is also affected by the dihedral symmetry of the square domain. We introduce a Corner-Difference MILP (CD-MILP) that represents each selected square by at most four signed corner coefficients. Two-dimensional prefix reconstruction shows that the resulting constraints are equivalent to the original cell-cover equations, while reducing the coverage-related nonzero count to $Θ(N^3)$. We also introduce lightweight corner-ordering constraints that retain at least one representative from every $D_4$ orbit, although ties may leave residual symmetry. Computational experiments on nine prime-size instances compare the baseline, CD-MILP, the baseline with corner ordering, and their combination using Gurobi and COPT. Under the stated experimental settings, the combined formulation solves eight instances with Gurobi and seven with COPT, compared with four and five, respectively, for the baseline. Its average presolved matrixs contain approximately $10^5$ nonzero coefficients on average, compared with approximately $10^7$ for the baseline.
